he "solves" by recalling a ready-made response. Almost any problem may come under the second type provided a person encounters it or very similar exercises sufficiently frequently so that the bond connecting the required response with the situation represented by the verbal statement of the problem has become fixed. When this happens the problem ceases to be a "problem" for the person in question, that is, it is not a situation requiring reflective thinking. For example, a seventh-grade pupil may think reflectively in solving an interest problem but a banker would respond to it in much the same way as the pupil responds to a request to multiply 846 by 52.

A verbal problem in the sense the term is used here is a new situation, that is, one for which the person does not have a ready-made response. Thus when we state that one of the objectives of instruction in arithmetic is to equip the pupils to solve verbal problems, we mean that they are to be equipped to respond satisfactorily to situations to which they have not responded previously, that is, to answer questions they have not answered in their study of arithmetic.

A new situation is not necessarily new in all its elements. In fact the opposite is usually true. A new problem will usually involve many familiar words and phrases. The implied questions relative to general quantitative relations will usually be familiar. The total situation, however, is new either because unfamiliar elements are introduced or because familiar elements appear in a new combination.

Response to problematic situations. As indicated in the preceding paragraphs, the nature of the response one makes is the distinguishing characteristic of a problematic situation. The response is complex. In so far as the situation is familiar, the elements of the response belong under other types of situations. Meanings are connected with words and symbols; the implied question concerning a functional relationship is identified and answered; denominate number facts are recalled; numbers are read and copied. However, the response cannot be adequately described by enumerating the responses to such elements. Reflective thinking is involved. It should be noted that the total response to a verbal problem includes the determination of the calculations to be performed plus the response to the example formulated. Reflective thinking is involved in only the first phase of the total response.

VII. Informational questions about business and social activities. Adults find occasion to answer a number of informational questions relating to such activities as banking, transportation, transmitting money, taxation, insurance, manufacturing, construction, and so forth. The following are typical: How is money transmitted? What is a promisory

note? What is a sight draft? How does a city secure funds for paving streets? How are taxes levied? What is board measure? What is overhead? How is postage computed on parcels? What conditions affect fire insurance rates? The questions may be asked in explicit terms but frequently they are implied in a general request or need.

The range of such questions for which instruction in arithmetic is expected to engender equipment has not been determined but it is obvious that other school subjects, especially geography and civics, must assume some of the responsibility for equipping pupils to answer informational questions relating to business and social activities.

Response given to informational questions relating to business and social activities. The general nature of the response to informational questions relating to business and social activities is implied by the illustrative questions in the preceding paragraph. However, it may be noted that usually precise and definite answers are required.

VIII. "Practical experiences." Under the head of "practical experiences" we group a number of types of situations such as (1) United States currency and other collections of objects to be counted, (2) magnitudes to be estimated or measured in terms of some unit, (3) business forms (sales slips, checks, money orders, etc.), catalogue lists, proposals for bond issues, newspaper quotations, and so forth to be comprehended, (4) situations in which arithmetical problems are to be identified and formulated.

The engendering of the arithmetical equipment required for responding to the situations enumerated under the head of "practical experiences" represents important objectives. The need for counting objects, estimating or measuring magnitudes, and comprehending business forms is generally recognized but the need for identifying and formulating the arithmetical problems arising in practical situations is even more important. With few exceptions adults seldom need to solve a verbal problem stated by another person. Their problems are encountered in their "practical experiences" and before the solution is begun the problem must be formulated, at least mentally.

General patterns of conduct as objectives in arithmetical instruction. As stated on page 10 a general pattern of conduct does not provide a response to a particular situation but it exercises a general control of one's responses to many situations, the range depending upon the extent of the generalization of the pattern.

Accuracy or the "habit of accuracy" is usually listed as an objective of arithmetical instruction but it is different from the specific habits

which function in performing calculations. The latter designate definite responses to definite situations. A "habit of accuracy" is a general pattern of conduct which controls responses to a variety of situations. which in this case are calculations. This control may result in performing the calculation a second time, checking, inspecting the work for errors, judging the answer with respect to reasonableness, and the like. A person who has attained the "habit of accuracy" tends to give one or more of these responses to any calculation situation. The presence of the word "habit" indicates that the response always tends to be made and is made skillfully. Much the same idea is expressed by the statement that a person who possesses the "habit of accuracy" knows what to do in order to attain accuracy and how to do it, and derives satisfaction from doing what is necessary.

Other general patterns of conduct listed among the objectives of arithmetic are neatness, honesty, initiative and resourcefulness in solving problems, perseverence, and systematic procedure. A general pattern of conduct which may be designated as a "problem solving attitude" is implied in some of the statements of the aim of arithmetic. Its central element appears to be the belief that the way to respond to a new situation, that is, a problem, is to ascertain what is known about it and precisely what question or questions are to be answered, and then to focus one's resources upon the problem in an attempt to manufacture a response by formulating solutions (hypotheses) and testing them until a satisfactory one is found. Usually there is coupled with this belief, confidence in one's own ability to solve the problem. The absence of these phases of a "problem solving attitude" is evidenced when a pupil searches in his text for the solution of a similar problem or restricts his efforts to recalling the solution of a similar problem.

Another significant phase of the "problem solving attitude" is involved in the pupil's concept of what it means to solve a problem. One point of view is that to solve a problem is to get the answer given in the text or one that will be accepted by the teacher. The "problem solving attitude" requires that one think of the solving of a problem as a case of reflective thinking in which the fundamental objective is to conform to the requirements of good thinking.

Summary. Although the preceding discussion of the objectives of arithmetic has filled several pages, it has doubtless been apparent to the reader that the items of mental equipment (specific habits, knowledge and general patterns of conduct) to be engendered by the instruction in

arithmetic have not been specified at all completely.20 However, the enumeration of eight types of situations to which pupils are to be equipped to respond and the consideration of general patterns of conduct should lead the reader to attach more meaning to the first statement of the purpose of instruction in arithmetic (see page 7). The teaching of arithmetic is expected to engender the specific habits, knowledge, and general patterns of conduct needed for responding in a satisfactory way to the following general classes of situations:

I. Number symbols.

II. Other arithmetical symbols and technical terms.

III. Two or more numbers quantitatively related with one missing.

IV. Examples.

V. Questions (usually implied) concerning functional relationships.

VI. Verbal problems.

VII. Informational questions about business and social activities. VIII. Practical experiences: (1) collections of objects to be counted, (2) magnitudes to be estimated in terms of some unit, (3) business forms, catalogue lists, newspaper quotations, and so forth to be comprehended, (4) situations in which arithmetical problems are to be identified and formulated.

The situations for which ability to respond should be engendered have been described only in terms of types and no attempt has been made to specify the quality of the several abilities.

CHAPTER II

THE PROCESSES OF LEARNING AND TEACHING

The discussion of the objectives of arithmetic in Chapter I furnishes a description of what pupils should learn during their study of this subject in the elementary school. The problem of this chapter is to describe certain phases of the learning process and the teaching procedures that are essential to the attainment of these objectives.

Learning an active process. In discussing the work of the teacher, we commonly use verbs such as "impart," "communicate," "present," "explain," and "instruct," which not infrequently appear to imply that in the process of educating children the teacher transmits specific habits, items of knowledge or general patterns of conduct to the pupil whose mind may be receptive or even eager to receive or may be indifferent or hostile. Although no one who is informed in regard to modern psychology would support such a theory of learning, the reading of current educational literature and the observation of classroom procedures suggest that many teachers, in planning lessons and in conducting recitations, do assume that their function is to transmit ideas, facts, rules, and other items of knowledge to their pupils.

The statement, "learning is an active process," is commonplace but its significance is far-reaching. What a child learns is the product of his own activity, physical, mental, and emotional. A child who is not active does not learn. In order to learn the multiplication combinations a child must enage in certain types of activity;1 in no other way can he acquire the necessary specific habits (fixed associations).

Assignment of exercises required as a basis for attaining arithmetical objectives. Acceptance of the thesis, that learning is an active. process and that the acquiring of certain abilities requires participation in certain types of learning activities, raises the question: What means must the teacher employ to secure the pupil activity that will lead to the attainment of the objectives described in Chapter I? A child learns as the result of his activity outside of the school such as playing games; doing errands for his parents, including the making of purchases; reading newspapers, magazines and books; constructing toys, playhouses

'The reader should not interpret this statement to mean that all pupils must gɔ through the same activities. The activities of one pupil engaged in learning the multiplication combinations may differ in certain respects from those of another pupil working toward the same end but the activities of the two pupils will have certain common characteristics. For example, in this case both will involve repetition.